Calculate arc length knowing its subtended chord and circumference diameter

Question

how do I calculate an arc length knowing only its subtended chord and the circumference diameter? I don't know the angle between OA and OB. Yesterday I did an experiment and calculated that the diameter / arc ratio is an exponential function which tends to 1 when lowering the numbers. My data: d / a 20 / 31.6 19.5 / 26.4 17.5 / 21.4 13.5 / 14.4 10 / 10.2 5 / 5 Thanks!

Answer 1

The triangle O-A-B is defined since we know AB and OA = OB = 0.5*CD. Calculate the angle using the Law of Cosines (see wikipedia):

Therefore

angle = arccos(((0.5*CD)^2 + (0.5*CD)^2 - AB^2) / (2*0.5*CD*0.5*CD))
angle = arccos((0.25*CD^2 - AB^2) / (0.5*CD^2))

Then you can compute the arc length using the angle

arc = pi * CD * angle / 360°

Answer 2

arc = CD asin(AB/CD)

(where the angle returned by asin is in radians)

EDIT:

I thought the derivation was obvious, but maybe I should explain it.
Mark the midpoint of AB, call it E.
sin(AOE) = (AB/2)/r where r=CD/2
angle = 2 asin(AB/CD)
arc = r angle = CD asin(AB/CD)